COMPARISON BETWEEN THE KALMAN AND THE NON-LINEAR LEAST-SQUARES ESTIMATORS IN LOW SIGNAL-TO-NOISE RATIO LIDAR INVERSION Francesc Rocadenbosch, Michaël Sicard, Adolfo Comerón, M. N. Md. Reba, and Adriano Camps Universitat Politècnica de Catalunya (UPC), Dep. of Signal Theory and Communications (TSC), Remote Sensing Lab (RSLAB), C/Jordi Girona, 1-3, D4-016, 08034 Barcelona, SPAIN. Email:
[email protected], Phone: +34 93-401-60-85, Fax: +34 93-401-72-32
1. ABSTRACT The comparatively recent application of low-power high-repetition-rate diode light sources to low-cost backscatter lidars as compared to classical high-energy low-repetition rate laser sources opens the research framework of low signal-tonoise ratio (SNR) inversion methods. Thus, micro-pulse lidars usually operate with energies in the 5-40 PJ range and repetition rates in the kHz region to achieve, e.g., 30-60-s time resolution and 30-to-75-m spatial resolution using photon counting detection. In comparison to classic 0.1-1-J energy, 10-50-Hz repetition-rate laser sources with the same resolutions, this represents a 40-80 dB reduction in the SNR (this figure being defined at the receiver’s voltage output). Independent inversion of the optical atmospheric parameters of interest, namely the aerosol extinction, the aerosol backscatter, and the lidar ratio can only be tackled by combining at least one elastic and one inelastic Raman channel, multiple zenith-angle elastic returns under the assumption of a homogeneously horizontally stratified atmosphere or by means of a HSRL (High Spectral Resolution Lidar). In the single-scatter elastic lidar equation, range-dependent inversion of the sought-after optical parameters requires both the introduction of “a priori” correlation hypotheses between the extinction and the backscatter profiles such as the assumption of a linear/power-law dependency between the optical parameters (Klett’s method, [1]) or the assumption of a range-dependent aerosol lidar ratio (Klett-Fernald-Sasano’s (KFS) method [2][3]), plus a boundary calibration. Besides, a temperature/pressure balloon-borne measurement (or a US-standard atmospheric model) is used to separate the molecular from the total optical components inverted by the KFS method. This paper focus on joint estimation of a range-independent lidar ratio and time evolution of the atmospheric backscatter profile by means of 1) adaptive inversion based on extended Kalman filtering (EKF) [4] and 2) non-linear least-squares estimation (NLSQ) [5], under moderate-to-low signal-to-noise ratios (typically below 20 dB). This preliminary comparative study follows a computer intensive simulation approach and assumes a moderately turbid atmosphere (i.e., a one component atmosphere) so that the aerosol component can be assumed dominant along the inversion range (the boundary layer). The paper discusses on the estimation problem in terms of data sufficiency, noise impact, and model parameters (in the case of the EKF), with reference to previously published works [6][7][8]. The former point of data sufficiency, i.e., the classical question of how to retrieve two unknowns (the backscatter and the lidar ratio in this case) from one single equation (the observable noisy lidar power), is solved here by introducing the concept of data decimation in the backscatter estimates. This means that the backscatter coefficient is estimated in less inversion cells than all the available power-measurement cells (observation cells), following a 1-to-M ratio, so as to guarantee data sufficiency. Consequently, the inversion resolution is reduced by the same factor M as compared to the (raw-data) observation resolution. Performance parameters taken into account are: errors in the sought-after optical estimates, tracking capability on the atmospheric backscatter time fluctuations, observable power de-noising capability, and initial user-error variance reduction. Whenever possible, the time-animated examples shown for both estimators are, in turn, compared with the single-profile time-averaged Klett’s solution (or its variants) as reference. In the case of Klett’s inversion, time/spatial averaging of the whole time-animated set of simulated input lidar records is necessary to boost the signal-to-noise ratio to suitable levels apt for inversion. Finally, a preliminary real data example is presented, where obviously the filter’s model is just a rough reasonable approximation to the true blind atmospheric one.
Keywords: lidar, inversion, Kalman filter, least-squares, control theory. 2. ACKNOWLEDGEMENTS The authors wish to acknowledge the following entities for partially supporting this research work and lidar systems developed at UPC: European Commission under the EARLINET-ASOS (EU Coordination Action) contract nº 025991 (RICA), and (EU Specific Support Action) contract nº 011863 (RIDS): “Technology development programme towards a European Extremely Large Telescope”; MCYT (Spanish Ministry of Science and Technology) and FEDER funds under the projects TEC2006-07850/TCM, Complementary Actions CGL2006-26149-E/CLI, CTM2006-27154-E/TECNO, and Special Action REN2002-12784-E; MITYC (Spanish Ministry of Industry, Tourism and Commerce) under the PROFIT project, CIT-020400-2005-56. MCYT is also thanked for the Ramón y Cajal position hold by Dr. M. Sicard, and Local Government of Catalonia (Generalitat de Catalunya/AGAUR) for Mr. Md. Reba’s predoctoral fellowship. We also thank the fruitful discussions about the problem provided by Dr. Gregori Vázquez, Dep. Signal Theory and Communications (TSC), Universitat Politècnica de Catalunya (UPC). 3. REFERENCES [1]
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